3C 264 is one of the few FRI radio galaxies with detected TeV emission. It is a low-luminosity active galactic nucleus (LLAGN) and is generally associated with a radiatively inefficient accretion flow (RIAF). Earlier multiwavelength studies suggest that the X-ray emission originates from a jet. However, the possibility that the RIAF can significantly contribute to the X-rays cannot be ruled out. In particular, hard X-ray emission ≳10 keV has never been detected, making it challenging to distinguish between X-ray models. Here we report a NuSTAR detection up to 25 keV from 3C 264. We also present subpixel deconvolved Chandra images to resolve jet emission down to ∼0.″2 from the center of the unresolved X-ray core. Together with a simultaneous Swift observation, we have constrained the dominant hard X-ray emission to be from its unresolved X-ray core, presumably in its quiescent state. We found evidence of a cutoff in the energy around 20 keV, indicating that at least some of the X-rays from the core can be attributed to the RIAF. The Comptonization model suggests an electron temperature of about 15 keV and an optical depth ranging between 4 and 7, following the universality of coronal properties of black hole accretion. The cutoff energy or electron temperature of 3C 264 is the lowest among those of other LLAGNs. The detected hard X-ray emission is at least an order of magnitude higher than that predicted by synchrotron self-Compton models introduced to explain γ-ray and TeV emission, suggesting that the synchrotron electrons might be accelerated to higher energies than previously thought.